# Find all functions !

• February 1st 2010, 01:00 PM
Perelman
Find all functions !
Hii !

Find all the functions $f$ : $\mathbb{R}$ $\longrightarrow$ $\mathbb{R}$ such as :

$f(xf(y))=f(xy)+x$
• February 1st 2010, 11:29 PM
red_dog
Let $y=0$. Then $f(xf(0))=f(0)+x$.

If $f(0)=0\Rightarrow f(0)=f(0)+x, \ \forall x\in\mathbb{R}\Rightarrow x=0, \ \forall x\in\mathbb{R}$, contradiction.

Then $f(0)\neq 0$. Let $f(0)=a, \ a\neq 0$.

Then $f(ax)=a+x, \ \forall x\in\mathbb{R}$.

Replace $x$ with $\frac{x}{a}$:

$f(x)=a+\frac{x}{a}$.